Proofs of Theorems and Glossary of Terms
Menu Theorem 4 Three angles in any triangle add up to 180°. Theorem 6 Each exterior angle of a triangle is equal to the sum of the two interior opposite angles Theorem 9 In a parallelogram opposite sides are equal and opposite angle are equal Theorem 19 The angle at the centre of the circle standing on a given arc is twice the angle at any point of the circle standing on the same arc. Theorem 14 Theorem of Pythagoras : In a right angle triangle, the square of the hypotenuse is the sum of the squares of the other two sides Just Click on the Proof Required Go to JC Constructions
Proof: 3 + 4 + 5 = Straight line 1 = 4 and 2 = 55 Alternate angles 3 + 1 + 2 = 1 + 2 + 3 = Q.E.D. 45 Given: Given:Triangle 12 3 Construction: Construction:Draw line through 3 parallel to the base Theorem 4: Three angles in any triangle add up to 180°C. To Prove: To Prove: 1 + 2 + 3 = Menu Constructions Quit Use mouse clicks to see proof
Theorem 6: Theorem 6: Each exterior angle of a triangle is equal to the sum of the two interior opposite angles To Prove: To Prove: 1 = 3 + 4 Proof: 1 + 2 = ………….. Straight line 2 + 3 + 4 = ………….. Theorem 2. 1 + 2 = 2 + 3 + 4 1 = 3 + 4 Q.E.D Menu Constructions Quit Use mouse clicks to see proof
Given: Given:Parallelogram abcd cb ad Construction: Construction: Draw the diagonal |ac| Theorem 9: In a parallelogram opposite sides are equal and opposite angle are equal To Prove: To Prove:|ab| = |cd| and |ad| = |bc| abc = adc and abc = adc Proof:In the triangle abc and the triangle adc 1 = 4 …….. Alternate angles |ac| = |ac| …… Common 2 = 3 ……… Alternate angles ……… ASA = ASA. The triangle abc is congruent to the triangle adc ……… ASA = ASA. |ab| = |cd| and |ad| = |bc| abc = adc and abc = adc Q.E.D Menu Constructions Quit Use mouse clicks to see proof
Given: Given:Triangle abc Proof:** Area of large sq. = area of small sq. + 4(area )) (a + b) 2 = c 2 + 4(½ab) a2 a2 + 2ab +b 2 = c 2 + 2ab a2 a2 + b2 b2 = c2c2 Q.E.D. a b c a b c a b c a b c Construction: Construction:Three right angled triangles as shown Theorem 14: Theorem of Pythagoras : In a right angle triangle, the square of the hypotenuse is the sum of the squares of the other two sides To Prove: To Prove:a 2 + b 2 = c 2 Menu Constructions Quit Use mouse clicks to see proof
Theorem 19: Theorem 19: The angle at the centre of the circle standing on a given arc is twice the angle at any point of the circle standing on the same arc. To Prove:| boc | = 2 | bac | To Prove: | boc | = 2 | bac | Construction: Construction: Join a to o and extend to r r Proof:In the triangle aob a b c o | oa| = | ob | …… Radii | 2 | = | 3 | …… Theorem 4 | 1 | = | 2 | + | 3 | …… Theorem 3 | 1 | = | 2 | + | 3 | …… Theorem 3 | 1 | = | 2 | + | 2 | | 1 | = 2| 2 | Similarly | 4 | = 2| 5 | | boc | = 2 | bac | Q.E.D Menu Constructions Quit Use mouse clicks to see proof